Quantum Entanglement of Anyonic Charges and Emergent Spacetime Geometry

Intrinsically topologically ordered phases can host anyons. Here, we take the view that entanglement between anyons can give rise to an emergent geometry resembling Anti-de Sitter (AdS) space. We analyze the entanglement structure of fractionalized anyons using mutual information and interpret the results within this emergent geometric framework. As a concrete example, we consider pairs of $e/2$-charged semions that arise from instanton configurations in a disordered zigzag graphene nanoribbon. These fractional charges, located on opposite zigzag edges, show long-range quantum entanglement despite being spatially separated. We analyze the scale dependence of their entanglement and embed the ribbon into an AdS-like bulk geometry. In this setup, the entanglement structure defines minimal surfaces in the bulk, providing a geometric view of the edge correlations. This gives a holographic picture of fractionalized degrees of freedom in quasi-one-dimensional systems and shows how quantum entanglement can generate emergent geometry even without conformal symmetry.

💡 Research Summary

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The paper “Quantum Entanglement of Anyonic Charges and Emergent Spacetime Geometry” investigates how quantum entanglement between fractionalized anyons can generate an emergent geometric description reminiscent of anti‑de Sitter (AdS) space, even in a non‑conformal, quasi‑one‑dimensional system. The authors focus on disordered zigzag graphene nanoribbons (ZGNRs) that, under the combined influence of on‑site Hubbard repulsion and random on‑site potentials, host instanton configurations. Each instanton binds a pair of e/2‑charged semions—one on each opposite edge of the ribbon. These semions are not independent particles; rather, they form long‑range entangled pairs whose mutual information (MI) remains sizable despite macroscopic separation.

The study proceeds in several stages. First, a Hubbard Hamiltonian with disorder is introduced and treated within a self‑consistent Hartree‑Fock mean‑field framework, which reproduces results from density‑matrix renormalization group (DMRG) and bosonization analyses. The disorder breaks chiral, mirror, and translational symmetries, allowing instanton formation that fractionalizes the edge charge from e to e/2. The authors then compute the MI between the two edge‑localized semions by extracting von Neumann entropies of the individual edge subsystems and of the combined system, using DMRG‑generated reduced density matrices. The MI exhibits a logarithmic decay with edge‑to‑edge distance L, but saturates to a large value at short distances, indicating robust non‑local correlations.

To translate these correlations into geometry, the authors adopt the “entanglement‑induced geometry” paradigm. They define an effective distance d(L)=−ξ ln I(L), where ξ is a tunable length scale, and map this distance onto the radial coordinate of a two‑dimensional hyperbolic metric ds²=(dz²+dx²)/z². In this picture, the physical ribbon length x corresponds to the boundary coordinate, while the emergent bulk coordinate z encodes the renormalization‑group scale of entanglement. The MI‑derived distances define geodesics in the bulk, and the minimal surface anchored on the two edge points reproduces the Ryu‑Takayanagi entropy formula. Consequently, the entanglement pattern of the semion pair is interpreted as a minimal surface (or “wormhole”) connecting the two boundaries of the emergent AdS‑like space, providing a concrete realization of the ER = EPR conjecture in a solid‑state context.

The authors further analyze the curvature‑entanglement relationship. Strong short‑range MI corresponds to a bulk region of high curvature (small AdS radius), while the rapid decay of MI at larger separations signals a flattening of the bulk geometry. By examining the spectrum of the Laplacian constructed from the MI‑derived distance matrix, they extract effective radial and temporal coordinates, showing that the bulk curvature varies smoothly with the entanglement scale. This quantitative link mirrors recent proposals that Einstein’s equations can emerge from the requirement of maximal entanglement in small regions.

Finally, the paper discusses experimental feasibility. Modern atom‑precise fabrication of ZGNRs enables direct imaging of edge charge distributions via scanning tunneling microscopy (STM) or atomic force microscopy (AFM). Quantum state tomography or measurement‑based entropy estimation techniques could be employed to reconstruct the reduced density matrices of the edge subsystems, allowing direct extraction of MI. Time‑resolved quench experiments could probe the dynamical response of the entanglement structure, offering a way to observe the “bulk” geometry’s evolution.

In summary, the work demonstrates three key points: (1) disorder‑driven instantons in ZGNRs produce e/2 semion pairs with long‑range quantum entanglement; (2) the mutual information of these pairs can be mapped onto an emergent AdS‑like bulk geometry, with minimal surfaces encoding the edge‑edge correlations; and (3) this emergent geometry arises without conformal symmetry, extending holographic ideas to a broader class of condensed‑matter systems. The study thus provides a concrete, experimentally accessible platform for exploring the interplay between quantum entanglement, topological order, and emergent spacetime geometry.